Controlled-source electromagnetic (“CSEM”) geophysical surveys use active (man-made) sources to generate electromagnetic fields to excite the earth, and deploy receiver instruments on the earth's surface, the seafloor, or inside boreholes to measure the resulting electric and magnetic fields, i.e., the earth's response to the source excitation. FIG. 1 illustrates the basic elements of an offshore CSEM survey. A vessel tows a submerged CSEM transmitter 11 over an area of sub-sea floor 13. The electric and magnetic fields measured by receivers 12 are then analyzed to determine the electrical resistivity of the earth structures (subsurface formations) beneath the surface or seafloor. This technology has been applied for onshore mineral exploration, oceanic tectonic studies, and offshore petroleum and mineral resource exploration.
Active electromagnetic source signals can be treated as a sum of sinusoidal signals (e.g., a square-wave signal made up of a fundamental frequency with odd harmonics). An example of such a source is the horizontal electric dipole used in much CSEM work. As the offset, i.e., the distance between such a dipole source 11 and the receivers 12 increases, the sinusoidal signal can decay significantly. Moreover, the far offsets are often critical for determining deep resistivity structures of interest. As a result, a need exists to obtain the best possible signal-to-noise ratio for this sinusoidal signal.
Typical processing methods to improve signal noise for this EM data involve breaking the data into time windows over which Fourier analysis or a similar method is used to calculate the amplitude and phase of selected frequency component(s). See, for example, Constable and Cox, “Marine controlled-source electromagnetic sounding 2. The PEGASUS Experiment,” Journal of Geophysical Research 101, 5519-5530 (1996). These windows cannot be too large because signal amplitude and relative phase may change substantially within the analysis window. Small windows, however, allow only minimal signal-to-noise ratio improvement. Current methods require a compromise between these two extremes.
Another problem with existing methods is that they don't take advantage of signal and noise correlations. Low-frequency magnetotelluric (“MT”) noises, in particular, are a significant problem for active source marine EM imaging because they can masquerade as signal. (MT noise is electromagnetic emissions from natural, not active, sources.) Correlations between different detectors could be used to help separate active-source signals from these noises. Other signal and noise correlations (e.g., signal correlations on the two horizontal components) are not optimally used in current approaches.
The Kalman filter algorithm has its origins in navigation positioning problems and is particularly suited to the class of tracking problems (Kalman, 1960). Originally published by Kalman in Trans. of the ASME—J. of Basic Engr., 35-45 (1960), much has been published since on modifications and applications of the basic Kalman filter as summarized, for example, by Brown in Introduction to Random Signal Analysis and Kalman Filtering, John Wiley & Sons, N.Y. (1983). A few of these modifications are of significance to some embodiments of the current invention.
The standard Kalman filter runs in one direction and filters data in this direction (or time) sequence. Therefore only previous data influences the filter result. An important modification due to Rauch, et al., gives an optimal treatment that uses the entire time record: Rauch, “Solutions to the linear smoothing problem,” IEEE Trans. On Auto. Control, AC-8, 371 (1963); and Rauch, et al., “Maximum likelihood estimates of linear dynamic systems,” AIAA J. 3, 1445 (1965). Szelag disclosed another algorithmic modification that allows the filter to track sinusoidal signals of a known frequency; see “A short term forecasting algorithm for trunk demand servicing,” The Bell System Technical Journal 61, 67-96 (1982). This was developed to track annual cycles in telephone trunk load values.
FIG. 2 is a flow chart illustrating the Kalman algorithm. Reference may be had to Brown's treatise, page 200, for more details.
La Scala, et al., disclose use of a known extended Kalman filter for tracking a time-varying frequency. (“Design of an extended Kalman filter frequency tracker,”IEEE Transactions on Signal Processing 44, No. 3, 739-742 (Mar., 1996)) The formulation assumes that the signal remains constant in amplitude. The particular Kalman algorithm used is therefore aimed at tracking a signal of unknown frequency where the frequency may undergo considerable change. Lagunas, et al., disclose an extended Kalman filter to track complex sinusoids in the presence of noise and frequency changes, such as Doppler shifts. (“High Order Learning in Termporal Reference Array Beamforming,” Signal Proc. VI, Theories and Applications, Elsevier Sci. Pub. B.V., pp. 1085-1088 (1992)) Like the La Scala method, Lagunas's algorithm is designed to track sinusoids of unknown frequency. Both methods will therefore be sub-optimal if applied to track a signal with constant or near constant frequency. Lagunas's method is able to also track amplitude changes, provided the changes are relatively small. Neither invention is aimed at processing electromagnetic survey data obtained using an electromagnetic source transmitting known waveforms at a known frequency. There is a need for a method for tracking large amplitude variations and small phase changes about a known sinusoid, using large windows, or even all, of the electromagnetic data. The present invention satisfies this need.